Why is Lorentz Transformation defined with one super and one sub index?

Question

I came across the Lorentz transformation in tensor form, usually written as $$\Lambda ^\mu _{\nu}$$ I understand that the first index usually corresponds to rows and the second to columns, and while I understand the difference between $\Lambda ^\mu _{\nu}$ and $\Lambda_\mu^\nu$, I don't understand the difference between $\Lambda ^\mu _{\nu}$ and $\Lambda ^{\mu\nu}$ (or the subscript version). What is the difference between writing both indices (either up or down) and both indices staggered?

Answer 1

One way to define Lorentz transformations is that they are the group of transformations that leave the Minkowski metric invariant \begin{equation} \Lambda^\alpha_{\ \ \mu} \Lambda^\beta_{\ \ \nu} \eta_{\alpha \beta} = \eta_{\mu\nu} \end{equation} To emphasize that $\eta$ has not changed after the transformation, we want to have $\eta_{\alpha\beta}$ with two lower indices on the left and $\eta_{\mu\nu}$ on the right. Then that forces the indices on $\Lambda$ to be "one up, one down."

There's nothing wrong with defining $\Lambda_{\mu\nu} \equiv \eta_{\mu\alpha} \Lambda^\alpha_{\ \ \nu}$. It is just that $\Lambda_{\mu\nu}$ is not the quantity that naturally appears in the definition above.

More formally, $\Lambda^\mu_{\ \ \nu}$ is an example of a coordinate transformation \begin{equation} \Lambda^\mu_{\ \ \nu} = \frac{\partial y^\mu}{\partial x^\nu} \end{equation} where \begin{equation} y^\mu = \Lambda^\mu_{\ \ \nu} x^\nu \end{equation} The "Jacobian matrix" $\frac{\partial y^\mu}{\partial x^\nu}$ that naturally appears in coordinate transformations generally has one upstairs and one downstairs index because it inherits the transformation properties of partial derivatives.

Answer 2

This actually boils down to conventions about notation, and can all be phrased in terms of basic linear algebra. Let $V$ be a vector space. A basis of $V$ is a set of linearly independent vectors which span $V$. By convention we denote these vectors by something like $e_\mu$ with the label indicating the vector down.

A general vector of $V$ is then a linear combination of the $e_\mu$. By convention we write the coefficients of the linear combinations with upper indices so that $v = v^\mu e_\mu$. And here we also use the Einstein convention in which the sum over repeated indices, one up and one down, is left implicit without summation sign.

Now consider a linear transformation $\Lambda:V\to V$. It maps a vector $v$ into $\Lambda(v)$. Since it is linear we can write $$\Lambda(v)=\Lambda(v^\nu e_\nu)=v^\nu \Lambda(e_\nu).$$

Finally, since $e_\mu$ is a basis, we can expand $\Lambda(e_\nu)$ into this basis. By convention we denote the coefficients of expansion as $$\Lambda(e_\nu)=\Lambda^\mu_{\phantom\mu\nu}e_\mu$$

so that $\Lambda(v)=\Lambda^\mu_{\phantom\mu\nu}v^\nu e_\mu$. At this point we can skip mentioning the basis explicitly all the time and talk about the action of $\Lambda$ on $v$ as $\Lambda^\mu_{\phantom\mu\nu}v^\nu$.

Up to now everything is basically introduction of nice notation. We then may notice that $\Lambda^\mu_{\phantom\mu\nu}v^\nu$ exactly matches the result of matrix multiplication where we multiply a matrix $[\Lambda]$ with entries $\Lambda^\mu_{\phantom\mu\nu}$ where the first index, $\mu$, indicates the rows, and the second index, $\nu$, the columns, by a column vector $v^\nu$. In particular, the index placement is important and we really should prefer $\Lambda^\mu_{\phantom\mu\nu}$ over $\Lambda^\mu_\nu$ as in the OP.

So, the reason why there is one upper index and one lower index is that, by the conventions we use, basis vectors have lower indices and components of vectors in the basis have upper indices. This is by no means mandatory, and you will find linear algebra texts, especially in pure math, where different conventions are used. The conventions I have specified here are the most used ones in Physics for example.

Finally you talk about $\Lambda^{\mu\nu}$. Well, one needs a definition for that, since by the conventions we have specified we only defined $\Lambda^\mu_{\phantom\mu\nu}$. It turns out that when your vector space $V$ has a metric $\eta_{\mu\nu}$ you can use it to raise and lower indices. This is essentially a mapping between the vector space $V$ and its dual space $V^\ast$, whose elements are called covectors.

The dual space $V^\ast$ comprises linear functions $\omega:V\to \mathbb{R}$. It is also a vector space and so it also has a basis. By convention, basis covectors are denoted with upper indices, for example, $\theta^\mu$. A general covector will then expand in terms of this basis with coefficients which by convention get lower indices, $\omega = \omega_\mu \theta^\mu$.

The mapping between $V$ and $V^\ast$ determined by the metric is the index lowering and raising. In particular, given a vector $v^\mu$ it defines $v_\mu = \eta_{\mu\nu} v^\nu$ and given a covector $\omega_\mu$ it defines $\omega^\mu = \eta^{\mu\nu}\omega_\nu$ where $\eta^{\mu\nu}$ is the inverse of $\eta_{\mu\nu}$.

Once you have this extra structure, the metric, given the linear operator $\Lambda^\mu_{\phantom\mu\nu}$ you can define $\Lambda^{\mu\nu}=\eta^{\nu\sigma}\Lambda^\mu_{\phantom\mu\sigma}$ and $\Lambda_{\mu\nu}=\eta_{\mu\sigma}\Lambda^\sigma_{\phantom\sigma\nu}$. So this is the difference in writing $\Lambda$ with all indices up or down: it is defined by an extra contraction with the metric, and so represents a different object, although one that encodes the same information.